A.1 Sets and maps

A setX is a collection of objects, which we call elements. We write x ∈ X if x is an element of X. We also write X = {x, y,…} to denote the elements of X. The empty set ∅ = {} is the unique set containing no elements. A set U is a subset of X, written U ⊆ X, if x ∈ U implies x ∈ X. A set U is a proper subset of X, written U ⊂ X, if U ⊆ X and U ≠ X. The unionX ∪ Y of two sets X and Y is the set of all elements in X or in Y (or in both), whereas the intersectionX ∩ Y is the set of all elements that are in both X and Y. If X ∩ Y = ∅ then XmeetsY (and Y meets X). The (set-theoretic) difference of two sets X and Y is the set X\Y = X − Y = {x ∈ X : x ∉ Y}. (This definition does not require that Y be contained in X.) The complement of U ⊆ X is Ū ≔ X − U.

Kirchhoff's rules

Electrical circuits provide an example of a simple physical system in which homology and cohomology play a small but important part. We consider only the simplest case of a resistive network comprising batteries and resistors obeying Ohm's law (Figure G.1), although by employing complex impedances the theory can be extended to steady state circuits that also contain capacitors and inductors. To solve the circuit (that is, to find the currents through, and voltages across, each element) we may appeal to Kirchhoff's rules:

1. (1) (junction condition) the sum of the currents into a node is zero;

2. (2) (loop law) the sum of the voltage gains around any closed loop is zero.

The junction condition embodies the law of charge conservation, which says that you cannot create or destroy net charge. The loop law embodies the law of energy conservation (because the work W done on a charge q by a potential difference ∆V is W = q∆V, so if you the law did not hold, a charge could flow around a closed loop and gain energy ad infinitum).

A graph model

We can model such a circuit by means of a graph, which is just a bunch of points (or nodes or vertices) joined by a bunch of lines (or edges). Topologically, a graph is the same thing as a one-dimensional simplicial complex. For simplicity, we will assume that the graph has only one connected component. So, start with a connected graph G with n nodes and m edges that models the topology of the circuit. Label the nodes, then orient each edge arbitrarily.

Suppose you are walking your dog Spot. You put the leash on Spot and take him to his favorite tree to do his business, but he sees a squirrel on the tree and takes off after it. The squirrel is really tired of being chased, and decides to teach Spot a lesson. So, instead of climbing back up the tree, he runs in a counterclockwise direction around the trunk with Spot not far behind. As Spot follows him around the tree, the leash gets wound around the tree k times (assuming you stay in place while he is running). At this point Spot gives up and sits down near your feet to bark sullenly at the squirrel. We say that the winding number of the leash is k. No matter how you try to move the leash, unless you cut it, it will remain wound around the tree k times. That is, its winding number is a homotopy invariant. This prosaic example generalizes to higher dimensions and has interesting mathematical and physical applications.

We start with the stack of records theorem, so-called because it reveals that all smooth maps from a compact manifold to another manifold of the same dimension look like a smooth covering by a stack of records. (See Figure 9.1.)

The definitions

Let M be a differentiable manifold. Crudely put, a vector bundle is just a collection, or bundle, of vector spaces, one for each point p of M, that vary smoothly as p varies. For example, if M is a differentiable manifold and if TpM is the tangent space to M at a point p then the union TM of all the TpM as p varies over M is a vector bundle called the tangent bundle of M. Similarly, the cotangent bundle T*pM of M is just the union of all the cotangent spaces T*pM as p varies over M.

Essentially, a vector bundle over M is a space E that looks locally like the Cartesian product of M with a vector space. As we are primarily concerned with the local properties of vector bundles, we do not lose much by limiting ourselves to product bundles. But, for those who insist on knowing all the gory details, the official definition is provided here. The beginner should skim over the next two paragraphs and revisit them only as needed.